Solving a Trigonometric Equation to Hit a Moving Target

[peanut648]

So the orginial question asked what angle to fire a gun to hit a moving target, and i got all the way to 40sinθcosθ-10sinθ-1 = 0, but we aren't supposed to use calculators so how would you solve this by hand. Thanks!

 

[tiny-tim]

Welcome to PF!

Hi peanut648! Welcome to PF!

Either use cosθ = √(1 - sin²θ), or write everything in terms of tan(θ/2).

 

[Architeuthis Dux]

I would suggest using the trigonometric identities and basic algebraic techniques to solve this equation by hand. First, we can rewrite the equation as 40sinθcosθ-10sinθ-1 = 0 as (40sinθ-10)(cosθ-1) = 0. This means that either 40sinθ-10 = 0 or cosθ-1 = 0.

To solve for θ, we can use the inverse trigonometric functions. For the first case, 40sinθ-10 = 0, we can use the inverse sine function to find the value of θ. This would give us sinθ = 10/40 = 1/4. Using a trigonometric table or basic trigonometric ratios, we can find that θ = sin^-1(1/4) = 14.48 degrees.

For the second case, cosθ-1 = 0, we can use the inverse cosine function to find the value of θ. This would give us cosθ = 1, which means that θ = cos^-1(1) = 0 degrees.

Therefore, the two solutions for θ are 14.48 degrees and 0 degrees. However, we must consider the practicality of these solutions. Since we are dealing with a moving target, it is highly unlikely that the gun would need to be fired at a 0 degree angle. Therefore, the more practical solution would be 14.48 degrees.

In conclusion, by using trigonometric identities and inverse trigonometric functions, we can solve the given equation by hand without the use of a calculator. However, it is important to also consider the practicality of the solutions in a real-world scenario.